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Washington (CNN)One of the founding members of the bipartisan "Gang of Six" on immigration says the group will exist no longer -- replaced by efforts to build a much larger coalition.

South Carolina Sen. Lindsey Graham, a Republican fresh off a victory in his efforts to pass a short-term government funding bill until February 8 with a commitment from the Senate majority leader to consider immigration on the Senate floor, said Monday that efforts going forward will be a new group.

"Oh yeah," Graham told reporters when asked if he was describing the end of the six-member talks.

"The Gang of Six started the process," Graham said. "That's all it was there to do. We need the Gang of 60. So the Gang of Six is going to be replaced by the Gang of 60."

It generally requires 60 senators to advance legislation in the Senate, and Republicans currently only hold a 51-49 majority.

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Graham joined Senate No. 2 Democrat Dick Durbin in building the bipartisan six-member talks, which included two other Republicans and two other Democrats. Their months-long negotiations on immigration resulted in a proposal they briefed to President Donald Trump -- who rejected it in vulgar terms.

The dismissal of that deal partly led to the short government shutdown over the weekend, as Democrats were joined by Graham and fellow member of the gang, Arizona Republican Sen. Jeff Flake, in rejecting a four-week continuing resolution and demanding progress on immigration. Graham and Flake spent the weekend shuttling between Majority Leader Mitch McConnell's and Minority Leader Chuck Schumer's offices trying to broker a deal to end the impasse.

Graham also joined roughly 20 other senators, split evenly by party, who spent the weekend meeting in the office of Maine Sen. Susan Collins and working to find a deal they could take to their respective leadership.

That group, Graham said, should form the basis of talks going forward -- which he said should include more than immigration and also deal with disaster relief, defense spending and overall budget negotiations.

Graham said he would also talk to conservative hardliners who rejected the Gang of Six bill, like Georgia Sen. David Perdue and Arkansas Sen. Tom Cotton, for their ideas.

"Let's get new senators working on building on what's out there and see if we can get 60 votes," Graham said. "The people who kind of were involved in helping the leaders reopen the government need to stay involved in finding solutions."

Graham said the effort begins "today."

"There's no taking a break here," he said, hoping for progress by the State of the Union next week.

And he had a piece of advice for Trump, with whom he has had a fraught relationship since the meeting at the White House where his deal was rejected.